Convergence of $\sum_{n=1}^{\infty}\frac{x}{[(n-1)x+1](nx+1)}$

Question

Consider,

$$\sum_{n=1}^{\infty}\frac{x}{[(n-1)x+1](nx+1)}$$

defined on $[0,1]$

Does this series converge point-wise and uniformly to a function on $[0,1]$?

How can I approach this problem? Since I have nothing to show for effort, I request you to give me some hints.

Thanks.